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Semilinear integro-differential equations with compact semigroups. (English) Zbl 0919.34055
The paper is concerned with the integro-differential equation $\frac{du}{dt}+Au(t)= f(t,u(t))+\int_{t_0}^ta(t-s)g(s,u(s))ds, \;0\leq t_0<T_0\leq\infty, \;u(t_0)=u_0, \tag{1}$ in a Banach space $$X$$ with $$-A$$ being the generator of a compact semigroup $$T(t)$$, $$t\geq 0$$; $$f,g: J\times U\to X$$, $$J=[t_0,T)$$, $$U$$ is an open set in $$X$$, $$a\in L^1(J)$$, $$u_0\in U$$. It extends known results to the case $$g\not=0$$. It is proved that if the nonlinear maps $$f$$ and $$g$$ are continuous and $$a$$ is locally integrable in $$J$$, then for every $$u_0\in X$$ the initial problem (1) has a local mild solution $$u(t)$$ (that is $$u(t)$$, $$t\in[t_0,t_1)$$ is a solution to a corresponding integral equation to (1) for some $$t_1<T_0$$). For the global case it is proved that if $$f,g$$ are continuous, map bounded subsets to bounded subsets and $$a$$ is locally integrable, then the initial problem (1) has a mild solution $$u(t)$$ on the maximal interval of existence $$[t_0,T_{\max})$$.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 47G20 Integro-differential operators 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 47D06 One-parameter semigroups and linear evolution equations
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